# A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where $[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining $$[H_1]\leq [H_2] \mbox{ iff } K_1\subseteq K_2 \mbox{ for some } K_1\in [H_1] \mbox{ and } K_2\in [H_2].$$We remark that ${\rm Iso}(G)$ is a lattice for many groups of small order (e.g. ${\rm Iso}(A_4)$ is the pentagon lattice $N_5$). Also, there are finite groups $G$ such that ${\rm Iso}(G)$ is not a lattice (e.g. $D_{12}$, $D_8\times\mathbb{Z}_4$, ... and so on). Therefore the following question is natural: $$\text{For which finite groups G is the poset {\rm Iso}(G) a lattice?}$$Note that such a group $G$ satisfies the following interesting property: "for every two distinct prime divisors $p$ and $q$ of $|G|$, either all subgroups of order $pq$ in $G$ are cyclic or all subgroups of order $pq$ in $G$ are non-abelian".